Point (1,-5) And The System Of Inequalities Y ≤ 3x+2, Y>-2x-3 - A Comprehensive Analysis

Jul 13, 2025 by ADMIN 89 views

Determine the Relationship Between the Point (1,-5) and the Given System of Inequalities

Introduction

In the realm of mathematics, understanding the relationship between points and systems of inequalities is a fundamental concept. This article delves into the process of determining whether a given point satisfies a system of inequalities, using both algebraic and graphical approaches. Specifically, we will explore the relationship between the point (1, -5) and the following system of inequalities:

y ≤ 3x + 2
y > -2x - 3

This analysis will provide a comprehensive understanding of how to assess the position of a point relative to the solution set of a system of inequalities. The algebraic method involves direct substitution of the point's coordinates into the inequalities, while the graphical method involves visualizing the inequalities on a coordinate plane and observing the point's location relative to the shaded regions. By combining these two approaches, we can gain a robust understanding of the point's relationship with the system of inequalities. The ability to solve such problems is crucial in various mathematical contexts, including linear programming, optimization problems, and the study of feasible regions. This article aims to provide a clear and concise explanation of the methods involved, making it accessible to students and enthusiasts alike. Understanding these concepts is not only essential for academic purposes but also has practical applications in fields such as economics, engineering, and computer science, where inequalities are used to model constraints and optimize solutions.

Algebraic Approach: Verifying the Point (1,-5)

The algebraic method offers a direct and precise way to determine if a point satisfies a system of inequalities. To achieve this, we substitute the coordinates of the point into each inequality and check if the resulting statements are true. In our case, the point is (1, -5), and the system of inequalities is:

y ≤ 3x + 2
y > -2x - 3

Let's begin by substituting the x and y values of the point (1, -5) into the first inequality, y ≤ 3x + 2. Replacing x with 1 and y with -5, we get:

-5 ≤ 3(1) + 2
-5 ≤ 3 + 2
-5 ≤ 5

The resulting statement, -5 ≤ 5, is true. This indicates that the point (1, -5) satisfies the first inequality. However, to fully confirm its relationship with the system, we must also verify the second inequality.

Now, let's substitute the same coordinates into the second inequality, y > -2x - 3. Replacing x with 1 and y with -5, we get:

-5 > -2(1) - 3
-5 > -2 - 3
-5 > -5

The resulting statement, -5 > -5, is false. This is because -5 is not strictly greater than -5; they are equal. For the point to satisfy the second inequality, the statement must be true.

Since the point (1, -5) satisfies the first inequality but not the second, we can conclude that it does not satisfy the entire system of inequalities. For a point to be a solution to a system of inequalities, it must satisfy all inequalities in the system. This algebraic approach provides a clear and definitive answer, but it is also helpful to visualize this relationship graphically, which we will explore in the next section. Understanding the algebraic method is crucial because it provides a foundation for solving more complex problems involving systems of inequalities. It also highlights the importance of careful substitution and evaluation of the resulting statements. This method is not only applicable in mathematics but also in various fields where constraints and conditions need to be checked, such as in computer programming and engineering design.

Graphical Approach: Visualizing the Inequalities and the Point

The graphical approach provides a visual representation of the system of inequalities, allowing us to see the region of the coordinate plane that satisfies all the inequalities simultaneously. This method is particularly helpful for understanding the relationship between the point (1, -5) and the system. The system of inequalities we are considering is:

y ≤ 3x + 2
y > -2x - 3

To graph these inequalities, we first treat them as equations and plot the corresponding lines. For the first inequality, y ≤ 3x + 2, we graph the line y = 3x + 2. This is a linear equation with a slope of 3 and a y-intercept of 2. Since the inequality includes “less than or equal to,” we draw a solid line to indicate that points on the line are also part of the solution.

For the second inequality, y > -2x - 3, we graph the line y = -2x - 3. This line has a slope of -2 and a y-intercept of -3. Since the inequality is “greater than” (not equal to), we draw a dashed line to indicate that points on the line are not part of the solution.

Next, we need to shade the regions that satisfy each inequality. For y ≤ 3x + 2, we shade the region below the solid line, as this is where the y-values are less than or equal to the values on the line. For y > -2x - 3, we shade the region above the dashed line, as this is where the y-values are greater than the values on the line.

The solution set for the system of inequalities is the region where the shaded areas of both inequalities overlap. This overlapping region represents all the points (x, y) that satisfy both inequalities simultaneously.

Now, let’s consider the point (1, -5). To determine its relationship with the system graphically, we plot this point on the same coordinate plane. By observing the graph, we can see that the point (1, -5) lies below the solid line y = 3x + 2, which means it satisfies the inequality y ≤ 3x + 2. However, the point (1, -5) lies on the dashed line y = -2x - 3, but not in the shaded region above it. This confirms that it does not satisfy the inequality y > -2x - 3 because the line itself is excluded from the solution due to the strict inequality. Since the point does not fall within the overlapping shaded region, it does not satisfy the system of inequalities. This graphical analysis aligns with our algebraic findings, providing a visual confirmation of our conclusion. The graphical approach is invaluable for gaining an intuitive understanding of systems of inequalities. It allows us to visualize the solution set and quickly assess whether a given point is part of that set. This method is particularly useful in applications where visual representations can aid in problem-solving, such as in optimization and resource allocation scenarios.

Conclusion: The Point (1,-5) and the System of Inequalities

In conclusion, through both the algebraic and graphical approaches, we have determined the relationship between the point (1, -5) and the given system of inequalities:

y ≤ 3x + 2
y > -2x - 3

Algebraically, we substituted the coordinates of the point (1, -5) into each inequality. We found that the point satisfies the first inequality, y ≤ 3x + 2, as -5 ≤ 3(1) + 2 simplifies to -5 ≤ 5, which is a true statement. However, when we substituted the coordinates into the second inequality, y > -2x - 3, we obtained -5 > -2(1) - 3, which simplifies to -5 > -5. This statement is false, as -5 is not greater than -5. Since the point must satisfy both inequalities to be a solution to the system, the point (1, -5) does not satisfy the system of inequalities.

Graphically, we plotted the lines corresponding to the inequalities and shaded the appropriate regions. The inequality y ≤ 3x + 2 is represented by a solid line and the region below it, while the inequality y > -2x - 3 is represented by a dashed line and the region above it. The overlapping shaded region represents the solution set for the system. When we plotted the point (1, -5) on the same coordinate plane, we observed that it lies on the dashed line y = -2x - 3, but not in the shaded region above it. While it does lie in the region satisfying y ≤ 3x + 2, the fact that it does not satisfy both inequalities confirms that it is not a solution to the system.

Therefore, based on both the algebraic and graphical analyses, we can definitively state that the point (1, -5) does not satisfy the given system of inequalities. This comprehensive approach highlights the importance of understanding both the algebraic and graphical methods for solving problems involving systems of inequalities. The algebraic method provides a precise way to check the inequalities, while the graphical method offers a visual understanding of the solution set and the point's position relative to it. Mastering these techniques is crucial for success in various mathematical contexts and practical applications. The ability to analyze the relationship between points and systems of inequalities is a valuable skill in fields ranging from mathematics and computer science to engineering and economics. Understanding these concepts allows for effective problem-solving and decision-making in various real-world scenarios.